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MultiprojectiveVarieties :: coneOfLines

coneOfLines -- cone of lines on a subvariety passing through a point

Synopsis

Description

In the example below we compute the cone of lines passing through the generic point of a smooth del Pezzo fourfold in $\mathbb{P}^7$.

i1 : K := frac(QQ[a,b,c,d,e]); t = gens ring PP_K^4; phi = rationalMap {minors(2,matrix{{t_0,t_1,t_2},{t_1,t_2,t_3}}) + t_4};

o3 : MultirationalMap (rational map from PP^4 to PP^7)
i4 : X = image phi;

o4 : ProjectiveVariety, 4-dimensional subvariety of PP^7
i5 : ideal X

             2                                     2
o5 = ideal (t  - t t  + t t , t t  - t t  + t t , t  - t t  + t t , t t  -
             5    4 6    2 7   4 5    3 6    1 7   4    3 5    0 7   2 4  
     ------------------------------------------------------------------------
     t t  + t t , t t  - t t  + t t )
      1 5    0 6   2 3    1 4    0 5

o5 : Ideal of frac(QQ[a..e])[t ..t ]
                              0   7
i6 : p := projectiveVariety minors(2,(vars K)||(vars ring PP_K^4))

o6 = point of coordinates [a/e, b/e, c/e, d/e, 1]

o6 : ProjectiveVariety, a point in PP^4
i7 : coneOfLines(X,phi p)

o7 = surface in PP^7 cut out by 6 hypersurfaces of degrees 1^3 2^3 

o7 : ProjectiveVariety, surface in PP^7
i8 : ideal oo

                                         2                                 
                 -d     2c     -b     - c  + b*d         -d     c     b    
o8 = ideal (t  + --t  + --t  + --t  + ----------t , t  + --t  + -t  + -t  +
             2    e 4    e 5    e 6        2     7   1    e 3   e 4   e 5  
                                          e                                
     ------------------------------------------------------------------------
                                                        2                   
     -a     - b*c + a*d         -c     2b     -a     - b  + a*c     2       
     --t  + -----------t , t  + --t  + --t  + --t  + ----------t , t  - t t 
      e 6         2     7   0    e 3    e 4    e 5        2     7   5    4 6
                 e                                       e                  
     ------------------------------------------------------------------------
                                  2                                        
       d       -2c       b       c  - b*d 2                d       -c      
     + -t t  + ---t t  + -t t  + --------t , t t  - t t  + -t t  + --t t  +
       e 4 7    e  5 7   e 6 7       2    7   4 5    3 6   e 3 7    e 4 7  
                                    e                                      
     ------------------------------------------------------------------------
                                                                        
     -b       a       b*c - a*d 2   2          c       -2b       a      
     --t t  + -t t  + ---------t , t  - t t  + -t t  + ---t t  + -t t  +
      e 5 7   e 6 7        2    7   4    3 5   e 3 7    e  4 7   e 5 7  
                          e                                             
     ------------------------------------------------------------------------
      2
     b  - a*c 2
     --------t )
         2    7
        e

o8 : Ideal of frac(QQ[a..e])[t ..t ]
                              0   7

See also

Ways to use coneOfLines :

For the programmer

The object coneOfLines is a method function.